limits of functions

(0.001 seconds)

1—10 of 167 matching pages

1: 4.31 Special Values and Limits
§4.31 Special Values and Limits
4.31.1 $\lim_{z\to 0}\frac{\sinh z}{z}=1,$
4.31.2 $\lim_{z\to 0}\frac{\tanh z}{z}=1,$
4.31.3 $\lim_{z\to 0}\frac{\cosh z-1}{z^{2}}=\frac{1}{2}.$
2: 4.4 Special Values and Limits
§4.4(iii) Limits
4.4.13 $\lim_{x\to\infty}x^{-a}\ln x=0,$ $\Re a>0$,
4.4.14 $\lim_{x\to 0}x^{a}\ln x=0,$ $\Re a>0$,
4.4.19 $\lim_{n\to\infty}\left(\left(\sum^{n}_{k=1}\frac{1}{k}\right)-\ln n\right)=% \gamma=0.57721\ 56649\ 01532\ 86060\dots,$
3: 4.17 Special Values and Limits
4.17.1 $\lim_{z\to 0}\frac{\sin z}{z}=1,$
4.17.2 $\lim_{z\to 0}\frac{\tan z}{z}=1.$
4.17.3 $\lim_{z\to 0}\frac{1-\cos z}{z^{2}}=\frac{1}{2}.$
5: 20.13 Physical Applications
In the singular limit $\Im\tau\rightarrow 0+$, the functions $\theta_{j}\left(z\middle|\tau\right)$, $j=1,2,3,4$, become integral kernels of Feynman path integrals (distribution-valued Green’s functions); see Schulman (1981, pp. 194–195). …
6: 29.5 Special Cases and Limiting Forms
§29.5 Special Cases and Limiting Forms
29.5.5 ${\lim_{k\to 1-}\frac{\mathit{Ec}^{m}_{\nu}\left(z,k^{2}\right)}{\mathit{Ec}^{m% }_{\nu}\left(0,k^{2}\right)}=\lim_{k\to 1-}\frac{\mathit{Es}^{m+1}_{\nu}\left(% z,k^{2}\right)}{\mathit{Es}^{m+1}_{\nu}\left(0,k^{2}\right)}}=\frac{1}{(\cosh z% )^{\mu}}F\left({\tfrac{1}{2}\mu-\tfrac{1}{2}\nu,\tfrac{1}{2}\mu+\tfrac{1}{2}% \nu+\tfrac{1}{2}\atop\tfrac{1}{2}};{\tanh}^{2}z\right),$ $m$ even,
29.5.6 $\lim_{k\to 1-}\frac{\mathit{Ec}^{m}_{\nu}\left(z,k^{2}\right)}{\left.\ifrac{% \mathrm{d}\mathit{Ec}^{m}_{\nu}\left(z,k^{2}\right)}{\mathrm{d}z}\right|_{z=0}% }=\lim_{k\to 1-}\frac{\mathit{Es}^{m+1}_{\nu}\left(z,k^{2}\right)}{\left.% \ifrac{\mathrm{d}\mathit{Es}^{m+1}_{\nu}\left(z,k^{2}\right)}{\mathrm{d}z}% \right|_{z=0}}=\frac{\tanh z}{(\cosh z)^{\mu}}F\left({\tfrac{1}{2}\mu-\tfrac{1% }{2}\nu+\tfrac{1}{2},\tfrac{1}{2}\mu+\tfrac{1}{2}\nu+1\atop\tfrac{3}{2}};{% \tanh}^{2}z\right),$ $m$ odd,
7: 18.11 Relations to Other Functions
Jacobi
18.11.5 $\lim_{n\to\infty}\frac{1}{n^{\alpha}}P^{(\alpha,\beta)}_{n}\left(1-\frac{z^{2}% }{2n^{2}}\right)=\lim_{n\to\infty}\frac{1}{n^{\alpha}}P^{(\alpha,\beta)}_{n}% \left(\cos\frac{z}{n}\right)=\frac{2^{\alpha}}{z^{\alpha}}J_{\alpha}\left(z% \right).$
10: 28.12 Definitions and Basic Properties
However, these functions are not the limiting values of $\operatorname{me}_{\pm\nu}\left(z,q\right)$ as $\nu\to n$ $(\neq 0)$. … Again, the limiting values of $\operatorname{ce}_{\nu}(z,q)$ and $\operatorname{se}_{\nu}(z,q)$ as $\nu\to n$ $(\neq 0)$ are not the functions $\operatorname{ce}_{n}\left(z,q\right)$ and $\operatorname{se}_{n}\left(z,q\right)$ defined in §28.2(vi). …